Question 1

Write out the first five terms of the sequence.
-$$a _ { n } = n - 6$$

Accepted Answer

A)  $- 24 , - 18 , - 12 , - 6,0$ 
B)  $- 5 , - 4 , - 3 , - 2 , - 1$ 
C)  $1,2,3,4,5$ 
D)  $- 6 , - 5 , - 4 , - 3 , - 2$ 
A)  $- 24 , - 18 , - 12 , - 6,0$ 
B)  $- 5 , - 4 , - 3 , - 2 , - 1$ 
C)  $1,2,3,4,5$ 
D)  $- 6 , - 5 , - 4 , - 3 , - 2$

Question 2

State an explicit rule for the nth term of the recursively-defined sequence.
-$\mathrm { a } _ { \mathrm { n } } = \mathrm { a } _ { \mathrm { n } - 1 } + 8 ; \mathrm { a } _ { 1 } = 1$

Accepted Answer

A)  $a _ { n } = 8 + 1 ( n - 1 )$ 
B)  $a _ { n } = 8 + 1 ( n )$ 
C)  $a _ { n } = 1 + 8 ( n )$ 
D)  $a _ { n } = 1 + 8 ( n - 1 )$ 
A)  $a _ { n } = 8 + 1 ( n - 1 )$ 
B)  $a _ { n } = 8 + 1 ( n )$ 
C)  $a _ { n } = 1 + 8 ( n )$ 
D)  $a _ { n } = 1 + 8 ( n - 1 )$

Question 3

Find the sum of the first n terms of the sequence.
-$25,31,37,43 , \ldots ; n = 10$

Accepted Answer

A)  395 
B)  520 
C)  550 
D)  605 
A)  395 
B)  520 
C)  550 
D)  605

Question 4

Determine whether the sequence converges or diverges. If it converges, give the limit.
-$$36 , - 6,1 , - \frac { 1 } { 6 } , \ldots$$

Accepted Answer

A)  Converges; $\frac { 216 } { 5 }$ 
B)  Converges; 6660 
C)  Diverges 
D)  Converges; $\frac { 216 } { 7 }$ 
A)  Converges; $\frac { 216 } { 5 }$ 
B)  Converges; 6660 
C)  Diverges 
D)  Converges; $\frac { 216 } { 7 }$

Question 5

Write the sum using summation notation, assuming the suggested pattern continues.
-$$9 + 11 + 13 + 15 + \ldots + ( 2 n + 1 ) \ldots$$

Accepted Answer

A)  $\sum _ { n = 5 } ^ { 20 } ( 2 n + 1 )$ 
B)  $\sum _ { n = 4 } ^ { 20 } ( 2 n + 1 )$ 
C)  $\sum _ { n = 5 } ^ { \infty } ( 2 n + 1 )$ 
D)  $\sum _ { n = 4 } ^ { \infty } ( 2 n + 1 )$ 
A)  $\sum _ { n = 5 } ^ { 20 } ( 2 n + 1 )$ 
B)  $\sum _ { n = 4 } ^ { 20 } ( 2 n + 1 )$ 
C)  $\sum _ { n = 5 } ^ { \infty } ( 2 n + 1 )$ 
D)  $\sum _ { n = 4 } ^ { \infty } ( 2 n + 1 )$

Question 6

Write the sum using summation notation, assuming the suggested pattern continues.
-$$1000 + 1331 + 1728 + 2197 + \ldots + n ^ { 3 } + \ldots$$

Accepted Answer

A)  $\sum _ { n = 11 } ^ { \infty } n ^ { 3 }$ 
B)  $\sum _ { n = 10 } ^ { \infty } ( n - 1 ) ^ { 3 }$ 
C)  $\sum _ { n = 10 } ^ { \infty } ( \mathrm { n } + 1 ) ^ { 3 }$ 
D)  $\sum _ { n = 10 } ^ { \infty } n ^ { 3 }$ 
A)  $\sum _ { n = 11 } ^ { \infty } n ^ { 3 }$ 
B)  $\sum _ { n = 10 } ^ { \infty } ( n - 1 ) ^ { 3 }$ 
C)  $\sum _ { n = 10 } ^ { \infty } ( \mathrm { n } + 1 ) ^ { 3 }$ 
D)  $\sum _ { n = 10 } ^ { \infty } n ^ { 3 }$

Question 7

Find the sum.
-$$\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 \right)$$

Accepted Answer

A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 30 n } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 6 n } { 6 }$ 
C)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 30 } { 6 }$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 30 n } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 6 n } { 6 }$ 
C)  $\frac { n ( n + 1 ) ( 2 n + 1 ) + 30 } { 6 }$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$

Question 8

Expand the binomial.
-$$\left( x ^ { - 3 } + 2 \right) ^ { 5 }$$

Accepted Answer

A)  $x ^ { - 15 } + 10 x ^ { - 12 } + 40 x ^ { - 9 } + 80 x ^ { - 6 } + 80 x ^ { - 3 } + 32$ 
B)  $x ^ { - 15 } + 5 x ^ { - 12 } + 10 x ^ { - 9 } + 10 x ^ { - 6 } + 5 x ^ { - 3 } + 1$ 
C)  $32 x^{ - 15}$ 
D)  $x ^ { - 15 } + 32$ 
A)  $x ^ { - 15 } + 10 x ^ { - 12 } + 40 x ^ { - 9 } + 80 x ^ { - 6 } + 80 x ^ { - 3 } + 32$ 
B)  $x ^ { - 15 } + 5 x ^ { - 12 } + 10 x ^ { - 9 } + 10 x ^ { - 6 } + 5 x ^ { - 3 } + 1$ 
C)  $32 x^{ - 15}$ 
D)  $x ^ { - 15 } + 32$

Question 9

Find the coefficient of the given term in the binomial expansion.
-$x ^ { 6 }$ term, $( x - 3 ) ^ { 15 }$

Accepted Answer

A)  $- 15,015$ 
B)  $98,513,415$ 
C)  $- 98,513,415$ 
D)  5005 
A)  $- 15,015$ 
B)  $98,513,415$ 
C)  $- 98,513,415$ 
D)  5005

Question 10

Solve.
-In how many ways can the letters in the word PAYMENT be arranged if the letters are taken 6 at a time?

Accepted Answer

A)  42 
B)  5040 
C)  7 
D)  2520 
A)  42 
B)  5040 
C)  7 
D)  2520

Question 11

Find the sum of the first n terms of the sequence.
-$- 1 , - 3 , - 9 , \ldots ; \mathrm { n } = 12$

Accepted Answer

A)  $- 265,720$ 
B)  797,161 
C)  $- 797,161$ 
D)  $\frac { 797,161 } { 2 }$ 
A)  $- 265,720$ 
B)  797,161 
C)  $- 797,161$ 
D)  $\frac { 797,161 } { 2 }$

Question 12

Use mathematical induction to prove the statement is true for all positive integers n.
-$$\left( 3 ^ { 2 } \right) ^ { n } = 3 ^ { 2 n }$$

Accepted Answer

We will use mathematical induction to pr

Question 13

Find an explicit rule for the nth term of the arithmetic sequence.
-$1,3,5,7 , \ldots$

Accepted Answer

A)  $a _ { n } = 1 + 2$ 
B)  $a _ { n } = 1 + 2 ( n + 1 )$ 
C)  $a _ { n } = 1 + 2 ( n )$ 
D)  $a _ { n } = 1 + 2 ( n - 1 )$ 
A)  $a _ { n } = 1 + 2$ 
B)  $a _ { n } = 1 + 2 ( n + 1 )$ 
C)  $a _ { n } = 1 + 2 ( n )$ 
D)  $a _ { n } = 1 + 2 ( n - 1 )$

Question 14

Use mathematical induction to prove the statement is true for all positive integers n.
-$$1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

Accepted Answer

To prove the given statement using mathe

Question 15

Find the sum.
-$\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 9 k + 20 \right)$

Accepted Answer

A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + \frac { 9 n ( n + 1 ) } { 2 } + 20 n$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + \frac { 9 n ( n + 1 ) } { 2 }$ 
C)  $\frac { 9 n ( n + 1 ) } { 2 } + 20 n$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + 20 n$ 
A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + \frac { 9 n ( n + 1 ) } { 2 } + 20 n$ 
B)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + \frac { 9 n ( n + 1 ) } { 2 }$ 
C)  $\frac { 9 n ( n + 1 ) } { 2 } + 20 n$ 
D)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } + 20 n$

Question 16

Express the rational number as a fraction of integers.
-$$0.0131131131 \ldots$$

Accepted Answer

A)  $\frac { 131 } { 9900 }$ 
B)  $\frac { 131 } { 999 }$ 
C)  $\frac { 131 } { 9990 }$ 
D)  $\frac { 131 } { 990 }$ 
A)  $\frac { 131 } { 9900 }$ 
B)  $\frac { 131 } { 999 }$ 
C)  $\frac { 131 } { 9990 }$ 
D)  $\frac { 131 } { 990 }$

Question 17

Find a recursive rule for the nth term of the sequence.
-$2 , - 8,32 , - 128 , \ldots$

Accepted Answer

A)  $a _ { n } = a _ { n - 1 } \cdot - 3$ 
B)  $a _ { n } = a _ { n - 1 } \cdot - 4$ 
C)  $a _ { n } = a _ { n - 1 } \cdot - 7$ 
D)  $a _ { n } = a _ { n - 1 } \cdot - 8$ 
A)  $a _ { n } = a _ { n - 1 } \cdot - 3$ 
B)  $a _ { n } = a _ { n - 1 } \cdot - 4$ 
C)  $a _ { n } = a _ { n - 1 } \cdot - 7$ 
D)  $a _ { n } = a _ { n - 1 } \cdot - 8$

Question 18

Evaluate.
-$\left( \begin{array} { l } 12 \ 12 \end{array} \right)$

Accepted Answer

A)  1 
B)  0 
C)  2 
D)  $479,001,600$ 
A)  1 
B)  0 
C)  2 
D)  $479,001,600$

Question 19

Evaluate.
-$\left( \begin{array} { c } 296 \ 2 \end{array} \right)$

Accepted Answer

A)  43660 
B)  294 
C)  $\frac { 296 ! } { 294 ! }$ 
D)  87,320 
A)  43660 
B)  294 
C)  $\frac { 296 ! } { 294 ! }$ 
D)  87,320

Question 20

Solve.
-A certain radioactive isotope has a half-life of 2 days. If one is to make a table showing the half-life decay of a sample of this isotope from 32 grams to 1 gram; list the time (in days, starting with $t = 0$ ) in the first column and the mass remaining (in grams) in the second column, which type of sequence is used in the first column and which type of sequence is used in the second column?

Accepted Answer

A)  Arithmetic in the first; geometric in the second 
B)  Geometric in the first; arithmetic in the second 
C)  Arithmetic in the first; arithmetic in the second 
D)  Geometric in the first; geometric in the second 
A)  Arithmetic in the first; geometric in the second 
B)  Geometric in the first; arithmetic in the second 
C)  Arithmetic in the first; arithmetic in the second 
D)  Geometric in the first; geometric in the second

Write out the first five terms of the sequence. - $a _ { n } = n - 6$

State an explicit rule for the nth term of the recursively-defined sequence. - $\mathrm { a } _ { \mathrm { n } } = \mathrm { a } _ { \mathrm { n } - 1 } + 8 ; \mathrm { a } _ { 1 } = 1$

Find the sum of the first n terms of the sequence. - $25,31,37,43 , \ldots ; n = 10$

Determine whether the sequence converges or diverges. If it converges, give the limit. - $36 , - 6,1 , - \frac { 1 } { 6 } , \ldots$

Write the sum using summation notation, assuming the suggested pattern continues. - $9 + 11 + 13 + 15 + \ldots + ( 2 n + 1 ) \ldots$

Write the sum using summation notation, assuming the suggested pattern continues. - $1000 + 1331 + 1728 + 2197 + \ldots + n ^ { 3 } + \ldots$

Find the sum. - $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 \right)$

Expand the binomial. - $\left( x ^ { - 3 } + 2 \right) ^ { 5 }$

Find the coefficient of the given term in the binomial expansion. - $x ^ { 6 }$ term, $( x - 3 ) ^ { 15 }$

Solve. -In how many ways can the letters in the word PAYMENT be arranged if the letters are taken 6 at a time?

Find the sum of the first n terms of the sequence. - $- 1 , - 3 , - 9 , \ldots ; \mathrm { n } = 12$

Use mathematical induction to prove the statement is true for all positive integers n. - $\left( 3 ^ { 2 } \right) ^ { n } = 3 ^ { 2 n }$

Find an explicit rule for the nth term of the arithmetic sequence. - $1,3,5,7 , \ldots$

Use mathematical induction to prove the statement is true for all positive integers n. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Find the sum. - $\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 9 k + 20 \right)$

Express the rational number as a fraction of integers. - $0.0131131131 \ldots$

Find a recursive rule for the nth term of the sequence. - $2 , - 8,32 , - 128 , \ldots$

Evaluate. - $\left( \begin{array} { l } 12 \\ 12 \end{array} \right)$

Evaluate. - $\left( \begin{array} { c } 296 \\ 2 \end{array} \right)$

Functions and Graphs

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An Introduction to Calculus: Limits, Derivatives, and Integrals

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Exam 9: Discrete Mathematics

Write out the first five terms of the sequence. - an=n−6a _ { n } = n - 6an​=n−6

State an explicit rule for the nth term of the recursively-defined sequence. - an=an−1+8;a1=1\mathrm { a } _ { \mathrm { n } } = \mathrm { a } _ { \mathrm { n } - 1 } + 8 ; \mathrm { a } _ { 1 } = 1an​=an−1​+8;a1​=1

Find the sum of the first n terms of the sequence. - 25,31,37,43,…;n=1025,31,37,43 , \ldots ; n = 1025,31,37,43,…;n=10

Determine whether the sequence converges or diverges. If it converges, give the limit. - 36,−6,1,−16,…36 , - 6,1 , - \frac { 1 } { 6 } , \ldots36,−6,1,−61​,…

Write the sum using summation notation, assuming the suggested pattern continues. - 9+11+13+15+…+(2n+1)…9 + 11 + 13 + 15 + \ldots + ( 2 n + 1 ) \ldots9+11+13+15+…+(2n+1)…

Write the sum using summation notation, assuming the suggested pattern continues. - 1000+1331+1728+2197+…+n3+…1000 + 1331 + 1728 + 2197 + \ldots + n ^ { 3 } + \ldots1000+1331+1728+2197+…+n3+…

Find the sum. - ∑k=1n(k2+5)\sum _ { k = 1 } ^ { n } \left( k ^ { 2 } + 5 \right)∑k=1n​(k2+5)

Expand the binomial. - (x−3+2)5\left( x ^ { - 3 } + 2 \right) ^ { 5 }(x−3+2)5

Find the coefficient of the given term in the binomial expansion. - x6x ^ { 6 }x6 term, (x−3)15( x - 3 ) ^ { 15 }(x−3)15